# Inverse of a larger matrix where the inverse of the submatrix is known

Let $$A, A^{-1} \in \mathbb{R}^{n \times n}$$ be known matrices. Suppose we have an invertible matrix $$B \in \mathbb{R}^{(n+1) \times (n+1)}$$ of the following form:

$$B = \begin{bmatrix} A & b\\ b^T & 1 \end{bmatrix}$$

where $$b$$ is a column vector and $$c$$ is a row vector. How can I calculate matrix $$B^{-1}$$ from known matrices $$A$$ and $$A^{-1}$$? Can the Sherman–Morrison formula be applied here? If so, how?

As far as I understand, it can be applied if some perturbation is made to $$A$$. However, the problem here is that $$B$$ has a different shape than $$A$$. Appending $$A$$ with zero entries in the beginning will not work either because the same matrix with zeros added in the bottom row and the right column is not necessarily nonsingular.

You know $$\begin{pmatrix} A & 0 \\ 0 & 1 \end{pmatrix}^{-1} = \begin{pmatrix} A^{-1} & 0 \\ 0 & 1 \end{pmatrix}$$, and from there you can make two successive rank-$$1$$ modifications, first adding $$b$$ along the last column, then $$c$$ along the last row. So, using Sherman–Morrison twice should work.
• Presumably it is the same thing, but starting with $\pmatrix{A&b\\0&1\\}^{-1}=\pmatrix{A^{-1}&-A^{-1}b\\0&1\\}$ is good. – Brendan McKay Jun 7 '20 at 4:37